In this paper, we introduce an implementation of the extended finite element method for fracture problems within the finite element software ABAQUS TM . User subroutine (UEL) in Abaqus is used to enable the incorporation of extended finite element capabilities. We provide details on the data input format together with the proposed user element subroutine, which constitutes the core of the finite element analysis; however, pre-processing tools that are necessary for an X-FEM implementation, but not directly related to Abaqus, are not provided. In addition to problems in linear elastic fracture mechanics, non-linear frictional contact analyses are also realized. Several numerical examples in fracture mechanics are presented to demonstrate the benefits of the proposed implementation.

SUMMARYA new stress recovery procedure that provides accurate estimations of the discretization error for linear elastic fracture mechanic problems analyzed with the extended finite element method (XFEM) is presented. The procedure is an adaptation of the superconvergent patch recovery (SPR) technique for the XFEM framework. It is based on three fundamental aspects: (a) the use of a singular +smooth stress field decomposition technique involving the use of different recovery methods for each field: standard SPR for the smooth field and reconstruction of the recovered singular field using the stress intensity factor K for the singular field; (b) direct calculation of smoothed stresses at integration points using conjoint polynomial enhancement; and (c) assembly of patches with elements intersected by the crack using different stress interpolation polynomials at each side of the crack. The method was validated by testing it on problems with an exact solution in mode I, mode II, and mixed mode and on a problem without analytical solution. The results obtained showed the accuracy of the proposed error estimator.

SUMMARYThe application of the extended finite element method (XFEM) to fracture mechanics problems enables one to obtain accurate solutions more efficiently than with the standard finite element method. A component can be modelled without the need to build a mesh that matches the crack geometry, and thus remeshing as the crack grows is unnecessary. In the XFEM approach, the interpolation on certain elements is enriched with functions that make it feasible to represent the crack tip asymptotic displacement fields by using a local partition of unity method. However, the enrichment is only partial in the blending elements connecting the enriched zone with the rest of the mesh, and consequently pathological terms appear in the interpolation, which lead to increased error. In this study we propose enhancing the blending elements by adding hierarchical shape functions where appropriate; this permits compensating for the unwanted terms in the interpolation. This technique is an extension of the study of Chessa et al. (Int. J. Numer. Meth. Engng. 2003; 57:1015-1038) to fracture mechanics problems. The numerical results show that the proposed enhancement always results in greater accuracy. Moreover, enhancing the blending elements makes it possible to recover the convergence rate that is decreased when the degrees of freedom gathering technique is used to improve the condition number of the stiffness matrix.

This work presents an analysis methodology based on the use of the Finite Element Method (FEM) nowadays considered one of the main numerical tools for solving Boundary Value Problems (BVPs). The proposed methodology, so-called cg-FEM (Cartesian grid FEM), has been implemented for fast and accurate numerical analysis of 2D linear elasticity problems. The traditional FEM uses geometry-conforming meshes; however, in cg-FEM the analysis mesh is not conformal to the geometry. This allows for defining very efficient mesh generation techniques and using a robust integration procedure, to accurately integrate the domain’s geometry. The hierarchical data structure used in cg-FEM together with the Cartesian meshes allow for trivial data sharing between similar entities. The cg-FEM methodology uses advanced recovery techniques to obtain an improved solution of the displacement and stress fields (for which a discretization error estimator in energy norm is available) that will be the output of the analysis. All this results in a substantial increase in accuracy and computational efficiency with respect to the standard FEM. cg-FEM has been applied in structural shape optimization showing robustness and computational efficiency in comparison with FEM solutions obtained with a commercial code, despite the fact that cg-FEM has been fully implemented in MATLAB.

WileyGonzález Albuixech, VF.; Giner Maravilla, E.; Tarancón Caro, JE.; Fuenmayor Fernández, FJ.; Gravouil, A. (2013). Convergence of domain integrals for stress intensity factor extraction in 2-D curved cracks problems with the extended finite element method. International Journal for Numerical Methods in Engineering. 94(8):740-757. doi:10.1002/nme.4478. This is the peer reviewed version of the following article: Int. J. Numer. Meth. Engng 2013; 94:740-757, which has been published in final form at Wiley Online Library (wileyonlinelibrary.com SUMMARYThe aim of this study is the analysis of the convergence rates achieved with domain energy integrals for the computation of the stress intensity factors (SIF) when solving 2-D curved crack problems with the extended finite element method (XFEM). Domain integrals, specially the J-integral and the interaction integral, are widely used for SIF extraction and provide high accurate estimations with finite element methods. The crack description in XFEM is usually realized using level sets. This allows to define a local basis associated with the crack geometry. In this work the effect of the level set local basis definition on the domain integral has been studied. The usual definition of the interaction integral involves hypotheses that are not fulfilled in generic curved crack problems and we introduce some modifications to improve the behavior in curved crack analyses. Despite the good accuracy of domain integrals, convergence rates are not always optimal and convergence to the exact solution cannot be assured for curved cracks. The lack of convergence is associated with the effect of the curvature on the definition of the auxiliary extraction fields. With our modified integral proposal, the optimal convergence rate is achieved by controlling the q-function and the size of the extraction domain.